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## TwoDimensional Heat Analysis Finite Element Method

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### Two-Dimensional Heat Analysis. Finite Element Method. 20 November 2002. Michelle Blunt ... to strain matrix: {g}=[B]{t} [B] is derivative of [N] Finite ... – PowerPoint PPT presentation

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Title: TwoDimensional Heat Analysis Finite Element Method

1
Two-Dimensional Heat AnalysisFinite Element
Method
• 20 November 2002
• Michelle Blunt
• Brian Coldwell

2
Two-Dimensional Heat Transfer
• Fundamental Concepts
• Solution Methods
• Heat Flux
• Mathematical
• Experimental
• Theoretical
• Finite Differences
• Finite Element Analysis

3
One-Dimensional Conduction

4
Two-Dimensional Conduction
5
Experimental Model
• Two-dimensional heat transfer plate from lab 6.
• Upper and left boundary conditions are set at
0oC lower and right conditions are constant at
80oC.

6
Theoretical ModelFinite Difference
7
Theoretical ModelFinite Element
The fundamental concept of FEM is that a
continuous function of a continuum (given domain
?) having infinite degrees of freedom is replaced
by a discrete model, approximated by a set of
piecewise continuous functions having a finite
degree of freedom.
8
Structural vs Heat Transfer
• Select element type
• Select element type
• Assume displacement function
• Assume temperature function
• Stress/strain relationships
• Temperature relationships
• Derive element stiffness
• Derive element conduction
• Assemble element equations
• Assemble element equations
• Solve nodal displacements
• Solve nodal temperatures
• Solve element forces
• Solve element gradient/flux

9
Finite Element 2-D Conduction
Select Element Type
• 1-d elements are lines
• 2-d elements are either triangles,
quadrilaterals, or a mixture as shown
• Label the nodes so that the difference between
two nodes on any element is minimized.

10
Finite Element 2-D Conduction
Assume (Choose) a Temperature Function
Assume a linear temperature function for each
element as
where u and v describe temperature gradients at
(xi,yi).
3 Nodes 1 Element 2 DOF x, y
11
Finite Element 2-D Conduction
Assume (Choose) a Temperature Function
12
Finite Element 2-D Conduction
Define Temperature Gradient Relationships
Analogous to strain matrix gBt B is
derivative of N
13
Finite Element 2-D Conduction
Derive Element Conduction Matrix and Equations
14
Finite Element 2-D Conduction
Derive Element Conduction Matrix and Equations
Stiffness matrix is general term for a matrix of
known coefficients being multiplied by unknown
degrees of freedom, i.e., displacement OR
temperature, etc. Thus, the element conduction
matrix is often referred to as the stiffness
matrix.
15
Finite Element 2-D Conduction
Assemble Element Equations, Apply BCs
From here on virtually the same as structural
approach. Heat flux boundary conditions already
accounted for in derivation. Just substitute
into above equation and solve for the following
Solve for Nodal Temperatures
Solve for Element Temperature Gradient Heat Flux
16
Algor How many elements?
Elements 9 Time 6s Nodes 16 Memory 0.239MB
17
Algor How many elements?
Elements 16 Time 6s Nodes 25 Memory 0.255MB
18
Algor How many elements?
Elements 49 Time 7s Nodes 64 Memory 0.326MB
19
Algor How many elements?
Elements 100 Time 7s Nodes 121 Memory
0.438MB
20
Algor How many elements?
Elements 324 Time 7s Nodes 361 Memory
0.910MB
21
Algor How many elements?
Elements 625 Time 9s Nodes 676 Memory
1.535MB
22
Algor How many elements?
Elements 3600 Time 15s Nodes 3721 Memory
7.684MB
23
Algor How many elements?
Automatic Mesh
Elements 334 Time 7s Nodes 371 Memory
0.930MB
24
Algor Results Options
25
Algor How many elements?
Smaller Elements
Fewer Elements
• Higher accuracy
• More time, memory
• Faster
• Less storage space

26
References
• Kreyszig, Erwin. Advanced Engineering
Mathematics, 8th ed.(1999)
• Chapters 8, 9
• Logan, Daryl L. A First Course in the Finite
Element Method Using Algor, 2nd ed.(2001)
• Chapters 13

27
Questions?
• Ha ha ha!!!
• Here comes your assignment